Title: Relative Group Cohomology and the Orbit Category
 
Authors: Semra Pamuk and Erg{\" u}n Yal{\c c}{\i}n

Status: to appear in Communications in Algebra

Address1: 
Department of Mathematics, 
Middle East Technical University, 
06531, Ankara, Turkey 

Email1: pasemra@metu.edu.tr 

Address2:
Department of Mathematics, 
Bilkent University,
06800 Bilkent, Ankara, Turkey

Email2: yalcine@fen.bilkent.edu.tr 

Abstract: Let $G$ be a finite group and $\cF$ be a family of subgroups of $G$ closed under conjugation and taking subgroups. We consider the question whether there exists a periodic relative $\cF$-projective resolution for $\bbZ$ when $\cF$ is the family of all subgroups $H \leq G$ with $\rk H \leq \rk G-1$. We answer this question negatively by calculating the relative group cohomology $\cF H^* (G, \bbF_2)$ where $G=\bbZ/2\times \bbZ /2$ and $\cF$ is the family of cyclic subgroups of $G$. To do this calculation we first observe that the relative group cohomology $\cF H^*(G, M)$ can be calculated using the ext-groups over the orbit category of $G$ restricted to the family $\cF$. In second part of the paper, we discuss the construction of a spectral sequence that converges to the cohomology of a group $G$ and whose horizontal line at $E_2$ page is isomorphic to the relative group cohomology of $G$.